[Irregular activity oscillations of rotary molecular motor. A simple kinetic model of F1-ATPase].

F1-ATPase is a catalytic portion of the rotary molecular motor, F1Fo-ATP synthase. Cooperative ATP hydrolysis at the three catalytic sites of the F1-ATPase is connected with rotation of the central gamma-subunit inside a cylinder made of three a subunits and three beta subunits. Various experimental works have shown that the gamma-subunit rotates with irregular dwells. A simple kinetic model of this paper explains dwells during rotation as a result of the deterministic chaos. It is shown that the deterministic chaos occurs under the rate constants close to the known experimental estimations. Time duration of dwells in the model are close to those observed experimentally. Our model explains the known irregular occupancy of catalytic sites of F1-ATPase by nucleotides.

[A simple kinetic model for dynein oscillatory activity].

A kinetic model for dynein, a molecular motor, complexed with microtubule fragments, is considered. The model explains the experimental observations of oscillatory movements in surprisingly simple axonemal fragments perfused by the ATP solution. The model explains at first time the oscillatory dynein activity as a phenomenon induced by two dynein heads cooperative interaction in the axoneme. The oscillation form, frequency, and amplitude, observed for the model, are close to these experimental characteristics. Kinetic parameters, used in the model, are close to the known experimental parameters.

[The inflow of the substrate can regulate the persistence of memory of enzymes having the properties of hysteresis: theoretical analysis].

A simple kinetic model of hysteretic enzymes with the influx of substrate or ion (transported by the enzyme) is considered. Two alternative steady activity levels are shown to arise in the system with a hysteretic enzyme. The transition between these levels can proceed in an oscillatory manner. The duration of the initial steady activity level is shown to be determined by the initial substrate (or ion) level, and the oscillatory transition between the activity levels is the property of hysteretic enzymes. It was shown for plasma membrane Ca2+-ATPase as an example that the level of the signal can be encoded into the time interval in which the enzyme retains the memory about this signal.

[Calmodulin can induce and control damping oscillations in the plasma membrane Ca2+ -ATPase activity: a kinetic model].

Plasma membrane Ca2+-ATPase is the calcium pump that extrudes calcium ions from cells using ATP hydrolisis for the maintenance of low Ca2+ concentrations in the cell. Calmodulin stimulates Ca2+-ATPase by binding to the autoinhibitory enzyme domain, which allows the access of cytoplasmic ATP and Ca2+ to the active and transport cites. Our kinetic model predicts damped oscillations in the enzyme activity and interprets the known nonmonotonous kinetic behavior of the enzyme in the presence of calmodulin. For the parameters close to the experimental ones, the kinetic model explains the changes in frequency and damping factor of the oscillatory enzyme activity, as dependent on calmodulin concentration. The calculated pre-steady-state curves fit well the known experimental data. The kinetic analysis allows us to assign Ca2+-ATPase to the hysteretic enzymes exhibiting activity oscillations in open systems.

[A model of an enzyme with mixed cooperation: 3 states of aspartate transcarbamylase]. / Model' fermenta so smeshannoi kooperativnost'iu: tri sostoianiia aspartat-transkarbamilazy.

Allosteric enzyme models on the basis of the known properties of aspartate transcarbamylase (ATCase) from Escherichia coli are suggested. In the first model molecules are supposed to equilibrate between two states. In contrast to the classical Monod-Wyman-Changeux model the symmetry of enzyme molecules changes during the conformational transition. It is shown that the number of binding sites of the enzyme defined from the Scatchard plots is sufficiently dependent on values of parameters of enzyme reaction. This fact results from the mixed (both positive and negative) cooperative effects. However the complex kinetic of ATCase is not completely simulated by this model. Therefore the model is complicated by taking into account the inactive third state of the enzyme. Thus the complex kinetic behaviour of ATCase is explained. The models may be also used for other enzymes.

[Enzymatic mechanisms of compensation of deleterious mutations]. / O fermentativnykh mekhanizmakh kompensatsii vrednykh mutatsii.

Mechanisms of stabilization and compensation, that occur in biochemical systems with enzymes modified by harmful mutations are considered. The compensation of such mutations can result in their evolutionary neutralism. The stabilization is considered due to kinetic signals of metabolites which form the direct and feedback connections with enzymes (temporal stabilization), and also the compensation in enzymatic aggregates determined by the changes of conformation (spatial stabilization). Examples of the stabilization in one or several steady states of enzymatic systems are presented. The neutralism of the distortion of inhibitory and catalytic properties of enzymes is shown in the region of stabilization of these properties.

[Simple kinetic models explaining critical phenomena in enzymatic reactions with isomerization of the enzyme and substrate]. / Prostye kineticheskie modeli, ob''iasniaiushchie kriticheskie iavleniia v fermentativnykh reaktsiiakh s izomerizatsiei fermenta i substrata.

Kinetic models for enzyme reactions are considered which take into account enzyme and substrate isomerization. Application of graph-theoretic methods allows to reveal fragments in schemes which may induce multiple stead-states or concentrational selfoscillations. The role of substrate isomers in the inhibition of enzyme isomers to produce critical phenomena is considered. The boundaries of parameter domains for critical phenomena are estimated. It is shown that the controlled change in concentrations of substrate and enzyme isomers may be important in regulation of enzyme systems, if different enzyme isomers are inhibited mainly by different substrate isomers. The models are used for interpretation of possible critical phenomena in the open reaction catalyzed by lactate dehydrogenase. It is shown that lactate dehydrogenase may act as a trigger in carbohydrate metabolism by changing "critically" its activity in relation to changes in pH and pyruvate fluxes. Slow enzyme inhibition by enolpyruvate is suggested as a possible reason for glycolytic oscillations.

[Symmetry-regulated dynamics of multi-enzyme complexes. A model of a pyruvate dehydrogenase complex from Escherichia coli]. / Upravliaemaia simmetriei dinamika mul'tifermentnykh kompleksov. Model' piruvatdegidrogenaznogo kompleksa iz Escherichia coli.

A dynamic model for quaternary structure of a multienzyme complex is considered. The model is based on the supposition of simultaneously existing similar subunits in a number of different conformational states in the "core" of the multienzyme complex. It is supposed that cyclic conformational transitions of the "core" subunits conserve the symmetry of the entire complex. Such transitions drive the core dynamics as well as the suprastructural multienzyme dynamics. The dynamic model is constructed for the pyruvate dehydrogenase complex from E. coli in a supposition of three different conformers existing in its "core" which correspond to the three steps of the cyclic catalytic process. The model is in accordance with the data from the literature.

[Structure-activity relationships in multienzyme complexes]. / Strukturno-funktsional'nye otnosheniia v mul'tifermentnykh kompleksakh.

Structures and molecular kinetic models of function of 2-oxo acid dehydrogenase complexes were analyzed. It was suggested that identical protein subunits in the multienzyme complexes as in the structure capable of self-assembly have identical contacts with the neighbors and identical environment. By sharing the enzyme aggregate subunits into distinct conformational classes the peripheral components were demonstrated to be arranged on the core so that the entire complex would have a definite symmetry. The number of the conformational classes is specified by the architecture of the core and considerations of symmetry. The results have allowed us to consider the mechanisms of functioning of the complexes. Specific examples are discussed.

[Analysis and interaction of myosin active centers during a mechanochemical cycle]. / Analiz vzaimodeistviia aktivnykh tsentrov miozina v protsesse mekhanokhimicheskogo tsikla.

A model of myosin complex with two active sites is considered. Each of the sites is supposed to pass through four states during a cycle of ATP hydrolysis. The model describes the distribution of the complex between states and the ATP hydrolysis rate as dependent on ATP concentration and reaction constants. Some of the constants were determined experimentally [6], but these values reproduce experimental curves [5] only with the assumption of cooperativity. If the first site is bind to actin, the rate of binding for the second one is shown to increase tenfold. If one of the sites is bind to actin by a "pulling" bridge, and the second site is in "rigor" state, then the ATP binding for the second site is about ten times faster. The transition into the rigor state proceeds much faster if both sites form pulling bridges. The rate of binding to actin is the same for one and for two sites.

[Observation of non-equilibrium phase transitions in a closed enzyme system. Intermittent regulation of the activity of lactate dehydrogenase and similar enzymes]. / O nabliudenii neravnovesnykh fazovykh perekhodov v zakrytoi fermentnoi sisteme. Skachkoobraznaia reguliatsiia aktivnosti laktatdegidrogenazy i skhodnykh fermentov.

Kinetic models of closed enzymic systems which allow the existence of multi-steady-states were considered. In the basis of these models lies the assumption about the coexistence in solution of different substrate forms and their non-enzymic transitions one to another. If the forms of substrate have different affinity to the enzyme and, moreover, one of the forms inhibits the enzyme, then the enzymic activity in certain conditions may change discretely. The transition of the system incidentally from one steady state to another is similar to the first order phase transition. It is shown that for the enzymic reaction catalyzed by lactate dehydrogenase, the conditions may exist for the non-equilibrium phase transition. It is shown that the existence of the metastability region in the system causes the unusual shifts of the discrete transition from one experiment to another. A model of an allosteric enzyme that describes the discrete behaviour similar to the phase transition is also considered.

[Symmetry of multienzyme complexes]. / O simmetrii polifermentnykh kompleksov.

A model for studying the symmetry of stable states arising from polyenzymic complex conformations is proposed. A formal scheme of submolecular structure self-assembly, on which the model is based, enables it not only to limit the class of conformations but in some cases to determine the structure of a complex in an unambigous manner. The model is shown in its application to polyenzymic complexes of dehydrogenases of alpha-keto acids.

[Regulation of two-substrate reaction by substrate analog, as analyzed by graphic methods]. / Regulirovanie dvukhsubstratnoi reaktsii analogom substrata, issledovannoe metodom grafovl.

The regulatory action of a substrate analogue on the two-substrate reaction has been analysed by the method of graphs. It was shown that the analogue can act not only as inhibitor but also as activator in an ordered mechanism of the two-substrate reaction. Activation of the reaction by the analogue is a consequence of its random mechanism in regard to substrates and products. This random mechanism tends to the appearance of another route in the enzymic reaction and by this route the activation of the reaction may be obtained. It was also shown, that substrate inhibition may be abolished by the analogue, if the reaction rate via analogue route is large enough. On this case the analogue competes with the substrate-inhibitor in the non-active ternary complex. For the analysis of kinetic schemes the cyclic algorithm of the graph method was applied because of its convenience for multiroute reactions. Some simplifications were proposed for the structure of graphs. The results of this analysis are applicable for interpretation of abolishing the substrate inhibition by monovalent anions for some dehydrogenases.

[pH-dependence of the structural and functional properties of lactate dehydrogenase (M4). Indirect cooperativity in lactate dehydrogenase]. / Issledovanie pH-zavisimosti strukturnykh i funktsional'nykh svoistv laktatdegidrogenazy (M4). O kosvennoi kooperativnosti v laktatdegidrogenaze.

The dependence of structural and functional properties of LDH on pH in the 6.0--9.0 region was investigated. There were no marked deviations of pyruvate reduction initial velocity curves from the Michaelis--Menten equation in a wide range of pyruvate concentrations. It was shown that Vmax changes negligibly in the 6.0--9.0 pH regions, but Km increased markedly with pH elevation. The pK value of 7.8+/-0.1 was obtained for 50% changes of pyruvate binding. The dependence of enzyme inhibition from pH at a high pyruvate concentration (20 mM) was investigated. At pH values above 8.0 pyruvate inhibition disappeared. The dependence of the inhibition degree from pH was estimated as pK 7.8+/-0.1. Hill coefficient (n) calculated from the curves of Km and the degree of substrate inhibition depending on pH was 1.6; n for pyruvate inhibition at pH 7.5 was 2 greater than n greater than 1 for moderate substrate concentrations (1--5 mM) and n approximately 1 for higher concentrations (5--40 mM). The value of n approximately 1 at pH 7.8 was obtained. The model suiting all available data concerning the cooperativity phenomena in LDH during protonation and inhibition by pyruvate is outlined. The model is based on the results indicating the slow isomerisation of LDH in ternary complexes with NADH and pyruvate and the absence of equilibrium on the intermediate stage of reaction.

[Kinetic manifestations of the interaction of active centers in swine skeletal muscle D-glyceraldehyde-3-phosphate dehydrogenase]. / Kineticheskie proiavleniia vzaimodeistvii aktivnykh tsentrov v D-glitseral'degid-3-fosfatdegidrogenaze iz skeletnykh myshts svin'i

The kinetic method and selective chemical modification have been used in studies of the kinetic manifestations of active site interactions in D-glyceraldehyde-3-phosphate dehydrogenase (GAP dehydrogenase). The reactions of glyceraldehyde and glyceraldehyde-3-phosphate oxidation were studied in the absence of substrate excess. In support of the data obtained previously it was shown that only a part of the tightly bound NAD molecules can be reduced after substrate addition. "Partial reducibility" is observed at various degrees of saturation of the enzyme with NAD involving a single NAD molecule per tetrametric enzyme. These facts can hardly be explained by assumption of functional non-equivalence of active sites, whether induced by coenzyme or preexisting in the apoenzyme. It was proven by selective alkylation of the catalytic SH groups that "partial reducibility" is due to the circumstance that equilibrium in the system under investigation is established at nearly equal NAD and NADH concentrations. A plot of initial reaction rates versus NAD concentration (at non-saturating substrate concentrations) gives S-shaped curves; this is explained by considerable enzyme activation upon saturation of the fourth site with coenzyme. After modification of three active sites with iodoacetate the S-shape of the curve disappeared. This fact leads to the conclusion that active site interactions are required for formation of the S-shaped curves. The activity of a single site functioning in the modified enzyme reached values equal to those of the active sites in the native enzyme in the fully activated state. A model is proposed which can explaine the variations in mode of enzyme activation in the native and modified states. It is suggested that the surroundings of all four SH groups must be altered in order to activate the enzyme; such changes can be induced either by alkylation of the SH groups or by NAD binding. Evidence is presented that important functional properties of GAP dehydrogenase cannot be elucidated at low enzyme concentrations and with excess of substrates: three active sites are saturated under such conditons and practically inactive, and the fourth site obeys Michaelis - Menten kinetics.

[Non-linear, non-competitive inhibition of lactate dehydrogenase with carboxylic acid anions (substrate analogues)]. / Nelineinoe nekonkurentnoe ingibirovanie laktatdegidrogenazy anionami karbonovykh kislot (analogami substrata).

The carboxylic acid anions are shown to inhibit lactate dehydrogenase in a non-competitive manner at pH 8.5. The inhibitory effect of different carboxylates used increases in the direction acetate leads to formate leads to methacrylate. Non-competitive inhibition by carboxylates has a non-linear character with effective Hill exponent depending on the inhibitor concentration increasing to the value n approximately equal to 2 when inhibitor concentration increases. The kinetic model is proposed to account for non-linear non-competitive inhibition in consequence of the abortive ternary complexes formation in two enzyme conformations. The substrate analogues properties that result in non-competitive inhibition of lactatedehydrogenase are considered.

[Ionophore-induced oscillations in erythrocytes. A kinetic model]. / Ionofor-indutsirovannye kolebaniia v éritrotsitakh. Kineticheskaia model'.

The kinetic model for K+, H+, Ca2+ concentrational self-oscillations in erythrocytes induced by A23187 and FCCP ionophores are considered. The model is based on the experimental data available and contains the minimal number of essential variables. The model was analysed by the method based on the graph representation of kinetic equations. The critical graph fragments provoking self-oscillatory trajectories in the system were revealed. It is shown that self-oscillatory behaviour is basically induced by conjugated processes produced by A23187. The parameter domain for self-oscillations is estimated including parameters of Ca2+-induced (through K+-channels) proton transport with FCCP participation. Numerical integration of kinetic equations was performed. The results obtained are in a good qualitative agreement with experimental data.

[A model of oligomeric enzymes explaining stepwise kinetic curves]. / Model' oligomernykh fermentov, ob''iasniaiushchaia "stupenchatye" kineticheskie krivye.

The complex kinetics of some oligomeric enzymes has been analysed. For interpreting such kinetics a theoretical model of an oligomeric enzyme is suggested, which is a modification of the "flip-flop" mechanism considered by Lazdunski et al. in 1971. Using our model, one can qualitatively interpret stepwise kinetic curves, i.e. the curves with few intermediate plateaus. Such curves are known from literature but have not exhaustively been explained. It is supposed that the enzyme kinetic curves can sometimes be of non-differential functions. The model used can also be applied to the kinetics of polyenzyme complexes.

[Kinetic manifestations of slow conformation rearrangements of lactate dehydrogenase. An experiment and a mathematical model]. / Issledovanie kineticheskikh proiavlenii medlennykh konformatsionnykh perestroek v laktatdegidrogenaze. Eksperiment i matematicheskaia model'.

Non-steady-state kinetics of lactate dehydrogenase (LDH) catalyzed reaction was investigated for a wide time interval (from 100 msec to 1-3 min) by using stopped-flow methods. A two-stage character of LDH reaction, slow changes like a lag-period on kinetic curves at pH 8.0, flexions on kinetic curves after pre-mixing LDH with NAD+ and pyruvate have been revealed. The graph theory for mathematical analysis of experimental data was applied, which has been developed for the non-steady-state kinetics. An enzyme model of the two-conformer LDH structure was used. The reaction scheme with a preferential inhibition of one of the conformers (pH 8.0) is suggested. The obtained values of kinetic constants prove that transitions between LDH conformers must be slow.